LOGARITHMS. 



to a greater is negative ; as this ratio, 

 compounded with any other ratio, dimin- 

 ishes it. The ratio of any quantity A to 

 unity, compounded with the ratio of unity 

 to A, produces the ratio of A to A, or the 

 ratio of equality ; and the measures of 

 those two ratios destroy each other when 

 added together ; so that when the one is 

 considered as positive the other is to be 

 coiibi.lered as negative. By supposing 

 tlic logarithms of quantities greater than 

 o a (which & supposed to represent unity) 

 to be positive, and the logarithms of quan- 

 tities less than it to be negative, the same 

 rules serve for the operations by loga- 

 rithms, whether tne^uantities be greater 

 or less than o a. When o p increases 

 proportionally, the motion of p is per- 

 petually accelerated ; for the spaces a c, 

 c f/, d e, &c. that are described by it in 

 any equal times that continually succeed 

 after each other, perpetually increase in 

 the same proportion as the lines o a, o c, 

 o d y &c. When the point p moves from a 

 towards o, and o p decreases proportion- 

 ally, the motion of p is perpetually re- 

 tarded ; for the spaces described by it in 

 any equal times that continually succeed 

 after each other, decrease in this case in 

 the same proportion as o p decreases. 



If the velocity of the point/? be always 

 as the distance o p, then will this line in- 

 crease or decrease in the manner sup- 

 posed by Lord Neper ; and the velocity 

 of the point p being the fluxion of the line 

 /, will always vary in the same ratio as 

 this quantity itself. This, we presume, 

 will give a clear idea of the genesis, or 

 nature of logarithms ; but for more of 

 this doctrine, see Maclaurin's Fluxions. 



LOGARITHMS, construction of. The first 

 makers of logarithms, had in this a very 

 laborious and difficult task to perform ; 

 they first made choice of their scale or 

 system of logarithms, that is, what set of 

 arithmetical progressionals should answer 

 to such a set of geometrical ones, for this 

 is entirely arbitrary ; and they chose the 

 decuple geometrical progressionals, 1, 10, 

 100, 1000, 10000, &c. and the arithmetical 

 one, 0, 1, 2, 3, 4, &c. or, 0.000000; 

 1.000000; 2.000000; 3.000000; 4.000000, 

 &c. as the most convenient. After this 

 they were to get the logarithms of all the 

 intermediate numbers between 1 and 10, 

 10 and 100,^100 and 1000, 1000 and 10000, 

 &c. But first of all they were to get the 

 logarithms of the prime numbers 3, 5, 7, 

 11, 13, 17, 19, 23, &c. and when these 

 were once had, it was easy to get those 

 of the compound numbers made up of the 

 prime ones, by the addition or subtraction 

 of their logarithm's. 



In order to this, they found a mean 

 proportion between 1 and 10, and its 

 logarithm will be one half that of 10 ; and 

 so given, then they found a mean pro- 

 portional between the number first found 

 and unity, which mean will be nearer to 

 1 than that before, and its logarithm will 

 be one half of the former logarithm, of 

 one-fourth of that of 10 ; and having in 

 this manner continually found a mean 

 proportional between 1 and the last mean, 

 and bisected the logarithms, they at 

 length, after finding 54 such means, came 

 to a number 



1.0000000000000001278191493200323442 

 so near to 1 as not to differ from it so 



much as Too^Tnrff^<JDTFoiTo?nKr part, and 

 found its logarithm to be 

 0.00000000000000005551II5I23I25782702 

 and 



00000000000000012781914932003235 to 

 be the difference whereby 1 exceeds the 

 number of roots or mean proportionals 

 found by extraction ; and then, by means 

 of these numbers, they found the loga- 

 rithms of any other numbers whatsoever; 

 and that after the following manner: be- 

 tween a given number, whose logarithm 

 is wanted, and 1, they found a mean pro- 

 portional, as above, until at length a num- 

 ber (mixed) be found, such a small mat- 

 ter above 1, as to have 1 and 15 cyphers 

 after it, which are followed by the same 

 number of significant figures ; then they 

 said, as the last number mentioned above, 

 is to the mean proportional thus found, 

 so is the logarithm above, viz. 

 0.00000000000000005551115123125782702 

 to the logarithm of the mean proportional 

 number, such a small matter exceeding 1, 

 as but now mentioned; and this logarithm 

 being as often doubled as the number of 

 mean proportionals, (formed to get that 

 number) willbe the logarithm of the given 

 number. And thi was the method Mr. 

 Briggs took to make the logarithms. But 

 if they are to be made to only seven 

 places of figures, which are enough for 

 common use, they had only occasion to 

 find 25 mean proportionals, or, which is the 



same thing, to extract the ' th 



root of 10. Now having the logarithms 

 of 3, 5, 7, they easily got those of 2, 4, 



6, 8, and 9; for since V= 2 the lo g a - 

 rithm of 2 will be the difference of the 

 logarithms of 10, and 5 the logarithm of 4 

 will be two times the logarithm of 2 ; the 

 logarithm of 6 will be the sum of the loga- 

 rithm of 2 and 3 ; and the logarithm of 9 

 double the logarithm of 3. So, also, hav- 

 ing found the logarithms of 13, 17, and 

 19, and alse of 23 and 29, they did easily 



